Wedelyn is a skiing technique first developed in Austria in the 1950s that consists of high speed turns made in succession with both skis parallel while not noticeably setting the ski edges on a slope.

According to Bob Barnes's "Complete Encyclopedia of Skiing (3rd Edition)":
"Wedelyn (or Wedel) means short, quick turns, generally involving a very narrow, often very narrow stance, with strong rotary movements and lots of skidding. Once thought of as the ultimate in expression of skiing skill . . . . "

Wedelyn is one of the 4 old school ski turns, others are: Serpentine; Mambo & Reuel

Wedelyn means "to wag" in German, was the ultimate heel-thrust skid. One have to bring one's feet sufficiently close together to do this. On steeper terrain, it became windshield wiper short swing. What carried over from Wedelyn to short swing was the quiet upper body moving down the fall line and the tails of the skis sliding across the snow.

Why Wedelyn was done wrong by so many skiers is that they took the word translation literally, tail wagging. So they set out to wag the tails by pushing them around from side to side, mostly by putting weight on them. Tail wagging is a result, not an intent of the manoeuvre. Correct Wedelyn is done with forward pressure and rotary input which makes the shovels bite and ALLOWS the LIGHTENED tails of the skis to brush out. Why Wedelyn was so revolutionary was that it was the first time turns, even quick ones, could be performed from the hips down without involving the upper body. Previously upper body input in the form of rotation or counter rotation was involved as turning forces.

For interest:
The Serpent turn is a down-motion turn off an edge-set with an anticipated upper body diving down the fall line. It was sort of the reverse of windshield wiper short swing in that the goal was more to turn the fronts of the skis into the fall line after the edge-set. The pivot point moved from the tips to under the feet.

The Mambo looks very slinky and is done by rotating the upper body as far as it will go, the movement then starts to to transfer to the skis which start to turn. At the moment the skis start to turn to the right, say, the upper body is rotated to the left as far as it is possible while the skis are still turning to the right and only will start to turn to the left when max rotation is archived. Repeat. It is a snake like movement.

Reuel is also called the Reuel Christy or Royal Christie, it is skiing the outside edge of the inside ski and hoisting the outside ski over your head.